Workshop on Quantum Fields and Nonlinear Phenomena, 27 September 2020. Organized by University of Craiova.

1. OBSERVATIONAL PARAMETERS OF
INFLATION IN HOLOGRAPHIC COSMOLOGY
MILAN MILOŠEVIĆ
Faculty of Sciences And Mathematics
University of Niš, Serbia
& SEENET-MTP Centre
Workshop on Quantum Fields and Nonlinear Phenomena
27 September 2020
in collaboration with: N. Bilić (Zagreb), G.S. Djordjević, D.D. Dimitrijević and M. Stojanović (Niš)
This work has been supported by the Serbian Ministry for Education, Science and Technological Development under the project No. 176021
and contract No. 451-03-68/2020-14/200124, as well as the ICTP - SEENET-MTP project NT-03 Cosmology-Classical and Quantum Challenges

2. INTRODUCTION
• The inflation theory proposes a period
of extremely rapid (exponential)
expansion of the universe during the an
early stage of evolution of the universe.
• The inflation theory predicts that during inflation (it takes about 10−34 𝑠) radius of the
universe increased, at least 𝑒60 ≈ 1026 times.
• Although inflationary cosmology has successfully complemented the Standard Model, the
process of inflation, in particular its origin, is still largely unknown.
• Recent years brought us a lot of evidence from WMAP and Planck observations of the CMB
• The most important way to test inflationary cosmological models is to compare the
computed and measured values of the observational parameters.
Figure: Baumann, D. TASI Lectures on Inflation. (2009), arXiv:0907.5424 [hep-th]

3. BRANEWORLD COSMOLOGY
• Braneworld universe is based on the scenario
in which matter is confined on a brane moving in the higher dimensional bulk with only gravity allowed
to propagate in the bulk.
• One of the simplest models - Randall-Sundrum (RS)
• RS model was originally proposed to solve the hierarchy problem (1999)
• Later it was realized that this model, as well as any similar braneworld model, may have interesting
cosmological implications
• Two branes with opposite tensions are placed at some distance in 5 dimensional space
• RS model – observer reside on the brane with negative tension, distance to the 2nd brane corresponds to the
Netwonian gravitational constant
• RSII model – observer is placed on the positive tension brane, the 2nd brane is pushed to infinity
N. Bilic, D.D. Dimitrijevic, G.S. Djordjevic, M. Milosevic, Tachyon inflation in an AdS braneworld with back-reaction, International Journal of Modern Physics A. 32 (2017) 1750039.
N. Bilic, G.B. Tupper, AdS braneworld with backreaction, Cent. Eur. J. Phys. 12 (2014) 147–159.

4. HOLOGRAPHIC BRANEWORLD
• Holographic braneworld - a cosmology
based on the effective four-dimensional
Einstein equations on the holographic
boundary in the framework of anti de
Sitter/conformal field theory (AdS/CFT)
correspondence.
• The model is based on a holographic braneworld scenario with an effective
tachyon field on a D3-brane located at the holographic boundary of an
asymptotic AdS5 bulk.
• The cosmology is governed by matter on the brane in addition to the
boundary CFT
time
Conformal
boundary at
z=0
space
z
xRSII brane
at z=zbr
N. Bilić, Randall-Sundrum vs Holographic Cosmology, IRB (2015)

5. HOLOGRAPHIC TACHYON COSMOLOGY
• The holographic braneworld is a spatially flat FRW universe with line element
൯𝑑𝑠2 = 𝑔 𝜇𝜈 𝑑𝑥 𝜇 𝑑𝑥 𝜈 = 𝑑𝑡2 − 𝑎2(𝑡)(𝑑𝑟2 + 𝑟2 𝑑𝛺2
• The holographic Friedmann equations
•
Where the scale 𝓁 can be identified with the AdS curvature radius and we introduced a
dimensionless expansion rate ℎ ≡ 𝓁𝐻 and the fundamental dimensionless coupling
ℎ2
−
𝓁2
4
ℎ4
=
𝜅2
3
𝓁4
𝜌
ሶℎ 1 −
𝓁2
2
ℎ2 = −
𝜅2
3
𝓁3(𝑝 + 𝜌)
Standard cosmology:
ℎ2
=
𝜅2
3
𝜌
ሶℎ = −
𝜅2
2
(𝑝 + 𝜌)
𝜅2 =
8𝜋𝐺 𝑁
𝓁2
Bilić, N., Dimitrijević, D. D., Djordjevic, G. S., Milošević, M. & Stojanović, M. Tachyon inflation in the
holographic braneworld. Journal of Cosmology and Astroparticle Physics 2019, 034–034 (2019).

6. HOLOGRAPHIC TACHYON COSMOLOGY
• Interesting property - solving the first Friedmann equation as a quadratic
equation
ℎ2
= 2 1 ± 1 −
𝜅2
3
𝓁4 𝜌
• We do not want our modified cosmology to depart too much from the standard
cosmology after the inflation era and demand that this equation reduces to the
standard Friedmann equation in the low density limit 𝜅2 𝓁4 𝜌 ≪ 1
• This demand will be met only by the (−) sign solution. We discard the (+) sign
solution as unphysical.

7. HOLOGRAPHIC TACHYON COSMOLOGY
• The physical range of the Hubble expansion rate is between ℎmin = 0 and the maximal value
ℎmax = 2
• It corresponds to the maximal energy density 𝜌max = Τ3 𝜅2
𝓁4
• Assuming no violation of the weak energy condition 𝑝 + 𝜌 ≥ 0, the expansion rate will be a
monotonously decreasing function of time.
• The universe starts from 𝑡 = 0 with an initial ℎ𝑖 ≤ ℎ 𝑚𝑎𝑥 with energy density and
cosmological scale both finite.
• The Big Bang singularity is avoided.
Bilić, N., Dimitrijević, D. D., Djordjevic, G. S., Milošević, M. & Stojanović, M. Tachyon inflation in the holographic braneworld. JCAP, 034–034 (2019).

8. EQUATIONS OF MOTION
• Tachyon matter in the holographic braneworld is described by the DBI Lagrangian
and the Hamiltonian
ℒ = −𝓁−4
𝑉( Τ𝜃 𝓁) 1 − 𝑔 𝜇𝜈 𝜃,𝜇 𝜃,𝜈
ℋ = 𝓁−4
𝑉 1 + 𝜂2
where 𝜂 =
𝑔 𝜇𝜈 𝜋 𝜇 𝜋 𝜈
𝓁4 𝑉
.
• As usual, the conjugate momentum is 𝜋 𝜇
=
𝜕ℒ
𝜕𝜃,𝜇
• The Hamilton equations are
𝜃,𝜇 =
𝜕ℋ
𝜕𝜋 𝜇 𝜋;𝜇
𝜇
= −
𝜕ℋ
𝜕𝜃

9. EQUATIONS OF MOTION
• The equations of motions
ሶ𝜃 =
𝜂
1 + 𝜂2
ሶ𝜂 = −
3ℎ𝜂
𝓁
−
𝑉,𝜃
𝑉
1 + 𝜂2 +
𝜂2
1 + 𝜂2
• As usual, the pressure and energy density are equal to Lagrangian and Hamiltonian
𝑝 ≡ ℒ = −𝓁−4
𝑉 1 − ሶ𝜃2 = −
𝓁−4 𝑉
1 − 𝜂2
𝜌 ≡ ℋ =
𝓁−4 𝑉
1 − ሶ𝜃2
= 𝓁−4
𝑉 1 − 𝜂2

10. INITIAL CONDITIONS
• Two natural initial conditions
a) 𝜂𝑖 = 0
b) ሶ𝜂𝑖 = 0
• The condition:
• (a) assures a finite initial ሶℎ, and
• (b) provides the solution consistent with the slow-roll regime

11. A) 𝜂𝑖 = 0
• 0 < ℎ𝑖 < 2
• From 𝑝 = −
𝓁−4 𝑉
1−𝜂2
and 𝜌 = 𝓁−4
𝑉 1 − 𝜂2 it follows 𝑝𝑖 = −𝜌𝑖
• From ℎ2 −
𝓁2
4
ℎ4 =
𝜅2
3
𝓁4 𝜌 and 𝜌 =
𝓁−4 𝑉
1− ሶ𝜃2
the initial 𝜃𝑖 can be fixed
𝑉 𝜃𝑖 =
3
𝜅2
ℎ𝑖
2
−
ℎ𝑖
4
4

12. B) ሶ𝜂𝑖 = 0
• From EqM we have
𝜂𝑖 = −
2 𝓁 Τ𝑉,𝜃 𝑉 𝑖
9ℎ 𝑖
2
−4 𝓁 Τ𝑉,𝜃 𝑉 𝑖
2
+3 9ℎ 𝑖
4
−4ℎ 𝑖
2
𝓁 Τ𝑉,𝜃 𝑉 𝑖
2
• From Friedmann equations we obtain
1 −
ℎ𝑖
2
2
2
= 1 −
𝜅2
3
𝑉 𝜃𝑖 1 + 𝜂𝑖
2
• Random numbers: ℎ𝑖, 𝜅 and parameters in the potential 𝑉 𝜃
• Numerical solutions: 𝜂𝑖 and 𝜃𝑖
• Not easy to solve, sometimes the real solution doesn’t exist, etc
𝑉 = 𝑉 𝜃
𝑉′𝜃 =
𝑑𝑉(𝜃)
𝑑𝜃

13. B) ሶ𝜂𝑖 = 0
• A different way
• Random numbers ℎ𝑖, 𝜃𝑖 and parameters in the potential 𝑉 𝜃
𝜅2
=
)3𝑉(𝜃𝑖
1 + 𝜂𝑖
2
1 − 1 −
ℎ𝑖
2
2
2
and calculate 𝜂𝑖 in the same way as in the previous way.

14. THE SLOW-ROLL PARAMETERS
• Number of e-folds
𝑁 𝑡 = න
𝑡 𝐶𝑀𝐵
𝑡 𝑒𝑛𝑑
𝐻 𝑡 𝑑𝑡
• Hubble hierarchy (slow-roll) parameters
𝜀𝑖+1 ≡
𝑑ln|𝜀𝑖|
𝑑𝑁
, 𝑖 ≥ 0, 𝜀0 ≡
𝐻∗
𝐻
where 𝐻∗ is the Hubble parameter at some chosen time
• The first two 𝜀 parameters 𝜀1 = −
ሶ𝐻
𝐻2, 𝜀2 = 2𝜀1 +
ሷ𝐻
𝐻 ሶ𝐻
, etc.
• The end of inflation 𝜀𝑖(𝜙 𝑒𝑛𝑑) ≈ 1
𝜙 𝑒𝑛𝑑 = 𝜙(𝑡 𝑒𝑛𝑑)

15. OBSERVATIONAL PARAMETERS
• Three independent observational parameters: amplitude of
scalar perturbation 𝐴 𝑠, tensor-to-scalar ratio 𝑟 and scalar
spectral index 𝑛 𝑠
𝑟 = 16𝜀1(𝜙𝑖)
𝑛 𝑠 = 1 − 2𝜀1(𝜙𝑖) − 𝜀2(𝜙𝑖)
• Satellite Planck (May 2009 – October 2013)
• Planck Collaboration
• The latest results were published in 2018.
At the lowest order in parameters 𝜀1 and 𝜀2
Planck 2018 results. X Constraints on inflation, arXiv:1807.06211 [astro-ph.CO]

16. OBSERVATIONAL PARAMETERS 𝑛 𝑠, 𝑟
𝑟 = 16𝜀1 1 + 𝐶𝜀2 +
)2(2 − ℎ2
)3(4 − ℎ2
𝑝𝑝,𝑋𝑋
𝑝,𝑋
2 𝜀1
𝑛s = 1 − 2𝜀1 − 𝜀2 − 2 +
8ℎ2
3 4 − ℎ2 2
𝑝𝑝,𝑋𝑋
𝑝,𝑋
2 𝜀1
2
− 3 + 2𝐶 +
)2(2 − ℎ2
)3(4 − ℎ2
𝑝𝑝,𝑋𝑋
𝑝,𝑋
2 𝜀1 𝜀2 − 𝐶𝜀2 𝜀3
• For 𝑋 = ሶ𝜃2 and 𝑝 = −𝑉 1 − 𝑋 we have
𝑝𝑝,𝑋𝑋
𝑝,𝑋
2 = −1
Bertini, N. R., Bilic, N. & Rodrigues, D. C. Primordial perturbations
and inflation in holographic cosmology. arXiv:2007.02332 [gr-qc]

17. POTENTIALS
𝑽 𝜽 =
𝟏
𝒄𝒐𝒔𝒉(𝝎𝜽)
𝑽 𝜽 = (𝟏 + 𝜽)𝒆−𝝎𝜽

18. 𝑽 𝜽 =
𝟏
𝒄𝒐𝒔𝒉(𝝎𝜽)

19. 𝑽 𝜽 =
𝟏
𝒄𝒐𝒔𝒉(𝝎𝜽)
60 < 𝑁 < 90
0 < 𝜔 < 0.25
0 < 𝜃𝑖 < 20
Colour represents the number of 𝑛 𝑠, 𝑟 points in a hexagon

20. 𝑽 𝜽 =
𝟏
𝒄𝒐𝒔𝒉(𝝎𝜽)

21. 𝑽 𝜽 =
𝟏
𝒄𝒐𝒔𝒉(𝝎𝜽)
- THE BEST FITTING RESULTS

23. 𝑽 𝜽 = (𝟏 + 𝜽)𝒆−𝝎𝜽

24. 𝑽 𝜽 = (𝟏 + 𝜽)𝒆−𝝎𝜽
60 < 𝑁 < 90
0 < 𝜔 < 0.25
0 < 𝜃𝑖 < 20

25. 𝑽 𝜽 = (𝟏 + 𝜽)𝒆−𝝎𝜽

26. 𝑽 𝜽 = (𝟏 + 𝜽)𝒆−𝝎𝜽
- THE BEST FITTING RESULTS

27. CONCLUSIONS
• We discussed a model of tachyon inflation based on a holographic braneworld
scenario with a brane located at the boundary of the AdS5 bulk.
• We simulated observational parameters of inflation for two potentials
𝑽 𝜽 =
𝟏
𝒄𝒐𝒔𝒉(𝝎𝜽)
, 𝑽 𝜽 = (𝟏 + 𝜽)𝒆−𝝎𝜽
• The agreement of our model with the Planck observational data is good,
especially for a higher number of e-folds.
• Preliminary results are promising and open good opportunity for further
analytical research of these potentials.
This work has been supported by the Serbian Ministry for Education, Science and Technological Development under the project No. 176021
and contract No. 451-03-68/2020-14/200124, as well as the ICTP - SEENET-MTP project NT-03 Cosmology-Classical and Quantum Challenges

28. REFERENCES
1. Bilić, N., Dimitrijević, D. D., Djordjevic, G. S., Milošević, M. & Stojanović, M. Tachyon inflation in the
holographic braneworld. Journal of Cosmology and Astroparticle Physics 2019, 034–034 (2019).
2. N. Bilic, D.D. Dimitrijevic, G.S. Djordjevic & M. Milosevic, Tachyon inflation in an AdS braneworld with
back-reaction, International Journal of Modern Physics A. 32 (2017) 1750039.
3. N.R. Bertini, N. Bilic & D.C. Rodrigues, Primordial perturbations and inflation in holographic
cosmology. arXiv:2007.02332 [gr-qc].
4. N. Bilić, Holographic cosmology and tachyon inflation. International Journal of Modern Physics A 33,
1845004 (2018).
5. N. Bilić, Randall-Sundrum versus holographic cosmology, Phys. Rev. D 93 (2016) 066010
[arXiv:1511.07323]